Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements

New vector finite elements are proposed for electromagnetics. The new elements are triangular or tetrahedral edge elements (tangential vector elements) of arbitrary polynomial order. They are hierarchal, so that different orders can be used together in the same mesh and p-adaption is possible. They provide separate representation of the gradient and rotational parts of the vector field. Explicit formulas are presented for generating the basis functions to arbitrary order. The basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning. Matrix assembly for the frequency-domain curl-curl equation is conveniently carried out by means of universal matrices. Application of the new elements to the solution of a parallel-plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient, for tetrahedral elements to order 4. In particular, the full-order elements have only the same asymptotic convergence rate as elements with a reduced gradient space (such as the Whitney element). However, further tests reveal that the optimum balance of the gradient and rotational components is problem-dependent.